Optimal control of a variational inequality with applications to structural analysis. I. Optimal design of a beam with unilateral supports

A class of optimal design problems is considered, where the state problem is governed by a variational inequality. The latter includes an elliptic operator, the coefficients of which are chosen as the design (control) variables.Existence of an optimal design is proven on the abstract level. Some applications are presented to the problems of elastic or elasto-plastic beams with unilateral supports. Finite element approximations are proposed and a theoretical convergence result is proven in case of elastic beams.